$title Minimal Surface with Obstacle COPS 2.0 #17 (MINSURF,SEQ=245)

$onText
Find the surface with minimal area, given boundary conditions, and
above an obstacle.

This model is from the COPS benchmarking suite.
See http://www-unix.mcs.anl.gov/~more/cops/.

The number of internal grid points can be specified using the command
line parameters --nx and --ny. COPS performance tests have been
reported for nx-1 = 50, ny-1 = 25, 50, 75, 100

Dolan, E D, and More, J J, Benchmarking Optimization
Software with COPS. Tech. rep., Mathematics and Computer
Science Division, 2000.

Friedman, A, Free Boundary Problems in Science and
Technology. Notices Amer. Math. Soc. 47 (2000),
854-861.

Keywords: nonlinear programming, engineering, minimal surface problem with obstacle
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$if not set nx $set nx 51
$if not set ny $set ny 26

Set
   nx 'grid points in 1st direction' / x0*x%nx% /
   ny 'grid points in 2st direction' / y0*y%ny% /;

Alias (nx,i), (ny,j);

Parameter
   hx   'grid spacing for x'
   hy   'grid spacing for y'
   area 'area of triangle';

hx   = 1/(card(nx) - 1);
hy   = 1/(card(ny) - 1);
area = 0.5*hx*hy;

Variable
   v(nx,ny) 'defines the finite element approximation'
   surf;

Positive Variable v;

Equation defsurf;

defsurf..
   surf/area =e= sum((nx(i+1),ny(j+1)), sqrt(1 + sqr((v[i+1,j] - v[i,j])/hx)
                                      + sqr((v[i,j+1] - v[i,j])/hy)))
              +  sum((nx(i-1),ny(j-1)), sqrt(1 + sqr((v[i-1,j] - v[i,j])/hx)
                                      + sqr((v[i,j-1] - v[i,j])/hy)));

v.fx['x0'   ,j] = 0;
v.fx['x%nx%',j] = 0;
v.fx[i,'y0'   ] = 1 - sqr(2*(ord(i) - 1)*hx - 1);
v.fx[i,'y%ny%'] = 1 - sqr(2*(ord(i) - 1)*hx - 1);

v.lo(i,j)$(((ord(i)-1) >= floor(0.25/hx) and (ord(i)-1) <= ceil(0.75/hx)) and
           ((ord(j)-1) >= floor(0.25/hy) and (ord(j)-1) <= ceil(0.75/hy))) = 1;

v.l(i,j) = 1 - sqr(2*(ord(i) - 1)*hx - 1);

Model minsurf / all /;

$if set workSpace minsurf.workSpace = %workSpace%

minsurf.workFactor = 2;

solve minsurf minimizing surf using nlp;